Question 13
For every positive integer $n$ determine the number of permutations $(a_{1}, a_{2}, \ldots, a_{n})$ of the set $\{1,2, \ldots, n\}$ with the following property:
$$
2(a_{1}+a_{2}+\cdots+a_{k}) \quad \text{is divisible by} \ k \quad \text{for} \ k=1,2, \ldots, n.
$$
$$
2(a_{1}+a_{2}+\cdots+a_{k}) \quad \text{is divisible by} \ k \quad \text{for} \ k=1,2, \ldots, n.
$$